<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation
Our Terms of Use (click here to view) and Privacy Policy (click here to view) have changed. By continuing to use this site, you are agreeing to our new Terms of Use and Privacy Policy.

Quadratic Inequalities

Graphs and sign tests for squared functions greater or less than a number.

Atoms Practice
Estimated6 minsto complete
%
Progress
Practice Quadratic Inequalities
Practice
Progress
Estimated6 minsto complete
%
Practice Now
Inequalities

Feel free to modify and personalize this study guide by clicking “Customize.”

Vocabulary

Compete the table.
Word Definition
Quadratic Inequality ___________________________________________________
___________ the values of that make equal to zero in a quadratic function
Polynomial Inequality: ___________________________________________________
___________ A function which may be expressed as a ratio of two polynomials, specified to be greater or less than a given value

Quadratic Inequalities

To solve quadratic inequalitites, you can graph it and see visually where the inequality is true. Without graphing, you can follow these steps:

  1. Set up the inequality in the form    (or  \begin{align*}p(x)<0, p(x)\le0,p(x)\ge0\end{align*} )
  2. Find the solutions to the equation \begin{align*}p(x)=0\end{align*} .
  3. Divide the number line into intervals based on the solutions to \begin{align*}p(x)=0\end{align*} .
  4. Use test points to find solution sets to the equation.

Example

 Find the solution set for the equation .

First, factor:             

\begin{align*}x^2 + 2x - 8 = 0\end{align*}

\begin{align*}(x + 4)(x - 2) = 0\end{align*}

Then create the intervals: 

 \begin{align*}(-\infty,-4) | (-4,2) | (2,\infty)\end{align*}

Finally, make a chart.

Interval Test Point Is \begin{align*}x^{2}+2x-8\end{align*} positive or negative? Part of Solution set?
\begin{align*}(-\infty,-4)\end{align*} \begin{align*}-5\end{align*} \begin{align*}+\end{align*} \begin{align*}yes\end{align*}
\begin{align*}(-4, 2)\end{align*} \begin{align*}1\end{align*} \begin{align*}-\end{align*} \begin{align*}no\end{align*}
\begin{align*}(2,+\infty)\end{align*} \begin{align*}3\end{align*} \begin{align*}+\end{align*} \begin{align*}yes\end{align*}

  if and only if \begin{align*}x < -4\end{align*} and  .

.

For more on quadratic inequalities, click here

Polynomial and Rational Inequalities

In polynomial and rational inequalities, the four basic steps are the same as in polynomial inequalities. For rational inequalities, however, you must also account for possible sign changes at vertical asymptotes or a break in the graph.

Add the vertical asymtotes to your intervals, and create the same chart as you did for quadratic inequalities. 

.

For more on polynomial and rational inequalities, click here


Practice

1) Find the solution set of the inequality \begin{align*}x^2 \leq 36\end{align*}

2) Find the solution set: 

3) Find the solution set: 

4) Graph the solution set: \begin{align*}(x - 3)(x + 4) \geq 0\end{align*}

.

Find the solution set of the following inequalities without using a calculator. Display the solution set on the number line.
  1. \begin{align*}x^{2}+2x-3\le0\end{align*}
  2. \begin{align*}-6x^{2}-13x+5\ge0\end{align*}
  3. \begin{align*}\frac{1-x}{x}<1\end{align*}
  4.  \begin{align*}\frac{x^4}{4} - x^2 <0\end{align*}
  5. \begin{align*}4x^3 - 8x^2 - x + 2 \geq 0 \end{align*}
Solve the following inequalities:
  1. \begin{align*}\frac{n^3 - 2n^2 - n + 2}{n^3 + 3n^2 + 4n + 12} < 0 \end{align*}
  2. \begin{align*}\frac{n^3 + 3n^2 - 4n - 12}{n^3 - 5n^2 + 4n - 20} \leq 0 \end{align*}
  3. \begin{align*}\frac{2n^3 + 5n^2 - 18n - 45}{3n^3 - n^2 +27n - 9} \geq 0 \end{align*}

My Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / notes
Show More

Image Attributions

Explore More

Sign in to explore more, including practice questions and solutions for Quadratic Inequalities.
Please wait...
Please wait...